Spin polarized states in neutron matter at a strong magnetic field
Spin polarized states in neutron matter at strong magnetic fields up to G are considered in the model with the Skyrme effective interaction. By analyzing the self-consistent equations at zero temperature, it is shown that a thermodynamically stable branch of solutions for the spin polarization parameter as a function of density corresponds to the negative spin polarization when the majority of neutron spins are oriented opposite to the direction of the magnetic field. Besides, beginning from some threshold density dependent on the magnetic field strength the self-consistent equations have also two other branches of solutions for the spin polarization parameter with the positive spin polarization. The free energy corresponding to one of these branches turns out to be very close to that of the thermodynamically preferable branch. As a consequence, at a strong magnetic field, the state with the positive spin polarization can be realized as a metastable state at the high density region in neutron matter which under decreasing density at some threshold density changes into a thermodynamically stable state with the negative spin polarization.
pacs:21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe
Neutron stars observed in nature are magnetized objects with the magnetic field strength at the surface in the range of - G LGS . For a special class of neutron stars such as soft gamma-ray repeaters and anomalous X-ray pulsars, the field strength can be much larger and is estimated to be about - G TD . These strongly magnetized objects are called magnetars DT and comprise about of the whole population of neutron stars K . However, in the interior of a magnetar the magnetic field strength may be even larger, reaching values of about G CBP ; BPL . The possibility of existence of such ultrastrong magnetic fields is not yet excluded, because what we can learn from the magnetar observations by their periods and spin-down rates, or by hydrogen spectral lines is only their surface fields. There is still no general consensus regarding the mechanism to generate such strong magnetic fields of magnetars, although different scenarios were suggested such as, e.g., a turbulent dynamo amplification mechanism in a neutron star with the rapidly rotating core at first moments after it goes supernova TD , or the possibility of spontaneous spin ordering in the dense quark core of a neutron star ST .
Under such circumstances, the issue of interest is the behavior of neutron star matter in a strong magnetic field CBP ; BPL ; CPL ; PG . In the recent study PG , neutron star matter was approximated by pure neutron matter in the model considerations with the effective Skyrme and Gogny forces. It has been shown that the behavior of the spin polarization of neutron matter in the high density region at a strong magnetic field crucially depends on whether neutron matter develops a spontaneous spin polarization (in the absence of a magnetic field) at several times nuclear matter saturation density as is usual for the Skyrme forces, or the appearance of a spontaneous polarization is not allowed at the relevant densities (or delayed to much higher densities), as in the case with the Gogny D1P force. In the former case, a ferromagnetic transition to a totally spin polarized state occurs while in the latter case a ferromagnetic transition is excluded at all relevant densities and the spin polarization remains quite low even in the high density region. Note that the issue of spontaneous appearance of spin polarized states in neutron and nuclear matter is a controversial one. On the one hand, the models with the Skyrme effective nucleon-nucleon (NN) interaction predict the occurrence of spontaneous spin instability in nuclear matter at densities in the range from to for different parametrizations of the NN potential R -RPV ( is the nuclear saturation density). For the Gogny effective interaction, a ferromagnetic transition in neutron matter occurs at densities larger than for the D1P parametrization and is not allowed for D1, D1S parametrizations LVRP . However, for the D1S Gogny force an antiferromagnetic phase transition happens in symmetric nuclear matter at the density IY2 . On the other hand, for the models with the realistic NN interaction, no sign of spontaneous spin instability has been found so far at any isospin asymmetry up to densities well above PGS -BB .
Here we study thermodynamic properties of spin polarized neutron matter at a strong magnetic field in the model with the Skyrme effective forces. As a framework for consideration, we choose a Fermi liquid approach for the description of nuclear matter AKPY ; AIP ; IY3 . Proceeding from the minimum principle for the thermodynamic potential, we get the self-consistent equations for the spin order parameter and chemical potential of neutrons. In the absence of a magnetic field, the self-consistent equations have two degenerate branches of solutions for the spin polarization parameter corresponding to the case, when the majority of neutron spins are oriented along, or opposite to the spin quantization axis (positive and negative spin polarization, respectively). In the presence of a magnetic field, these branches are modified differently. A thermodynamically stable branch corresponds to the state with the majority of neutron spins aligned opposite to the magnetic field. At a strong magnetic filed, the branch corresponding to the positive spin polarization splits into two branches with the positive spin polarization as well. The last solutions were missed in the study of Ref. PG . We perform a thermodynamic analysis based on the comparison of the respective free energies and arrive at the conclusion about the possibility of the formation of metastable states in neutron matter with the majority of neutron spins directed along the strong magnetic field. The appearance of such metastable states can be possible due to the strong spin-dependent medium correlations in neutron matter with the Skyrme forces at high densities.
Note that we consider thermodynamic properties of spin polarized states in neutron matter at a strong magnetic field up to the high density region relevant for astrophysics. Nevertheless, we take into account the nucleon degrees of freedom only, although other degrees of freedom, such as pions, hyperons, kaons, or quarks could be important at such high densities.
Ii Basic equations
The normal (nonsuperfluid) states of neutron matter are described by the normal distribution function of neutrons , where , is momentum, is the projection of spin on the third axis, and is the density matrix of the system I ; IY . Further it will be assumed that the third axis is directed along the external magnetic field . The energy of the system is specified as a functional of the distribution function , , and determines the single particle energy
Here the quantities and are matrices in the space of variables, with , , and being the Lagrange multipliers, being the chemical potential of neutrons, and the temperature.
Given the possibility for alignment of neutron spins along or opposite to the magnetic field , the normal distribution function of neutrons and single particle energy can be expanded in the Pauli matrices in spin space
As follows from the structure of the distribution functions , the quantity , being the exponent in the Fermi distribution function , plays the role of the quasiparticle spectrum. The spectrum is twofold split due to the spin dependence of the single particle energy in Eq. (3). The branches correspond to neutrons with spin up and spin down.
The distribution functions should satisfy the normalization conditions
Here is the total density of neutron matter, and are the neutron number densities with spin up and spin down, respectively. The quantity may be regarded as the neutron spin order parameter. It determines the magnetization of the system , being the neutron magnetic moment. The magnetization may contribute to the internal magnetic field . However, we will assume, analogously to Refs. PG ; BPL , that the contribution of the magnetization to the magnetic field remains small for all relevant densities and magnetic field strengths, and, hence,
This assumption holds true due to the tiny value of the neutron magnetic moment MeV/G A ( being the nuclear magneton) and is confirmed numerically by finding solutions of the self-consistent equations in two approximations, corresponding to preserving and neglecting the contribution of the magnetization.
Here is the free single particle spectrum, is the bare mass of a neutron, are the normal Fermi liquid (FL) amplitudes, and are the FL corrections to the free single particle spectrum. Note that in this study we will not be interested in the total energy density and pressure in the interior of a neutron star. By this reason, the field contribution , being the energy of the magnetic field in the absence of matter, can be omitted. Using Eqs. (1) and (9), we get the self-consistent equations in the form
To obtain numerical results, we utilize the effective Skyrme interaction. The amplitude of NN interaction for the Skyrme effective forces reads VB
where is the spin exchange operator, and are some phenomenological parameters specifying a given parametrization of the Skyrme interaction. In Eq. (12), the spin-orbit term irrelevant for a uniform matter was omitted. The normal FL amplitudes can be expressed in terms of the Skyrme force parameters AIP ; IY3 :
Further we do not take into account the effective tensor forces, which lead to coupling of the momentum and spin degrees of freedom HJ ; D ; FMS , and, correspondingly, to anisotropy in the momentum dependence of FL amplitudes with respect to the spin quantization axis. Then
where the effective neutron mass is defined by the formula
where is the effective mass of a neutron with spin up () and spin down ()
Note that for totally spin polarized neutron matter
where is the effective neutron mass in the fully polarized state. Since usually for Skyrme parametrizations , we have the constraint , which guarantees the stability of totally polarized neutron matter at high densities.
It follows from Eq. (19) that the effective chemical potential for neutrons with spin-up () and spin-down () can be determined as
Iii Solutions of self-consistent equations at . Thermodynamic stability
Here we directly solve the self-consistent equations at zero temperature and present the neutron spin order parameter as a function of density and magnetic field strength. In solving numerically the self-consistent equations, we utilize SLy4 and SLy7 Skyrme forces CBH , which were constrained originally to reproduce the results of microscopic neutron matter calculations (pressure versus density curve). Note that the density dependence of the nuclear symmetry energy, calculated with these Skyrme interactions, gives the neutron star models in a broad agreement with the observables such as the minimum rotation period, gravitational mass-radius relation, the binding energy, released in supernova collapse, etc. RMK . Besides, these Skyrme parametrizations satisfy the constraint , obtained from Eq. (21).
We consider magnetic fields up to the values allowed by the scalar virial theorem. For a neutron star with the mass and radius , equating the magnetic field energy with the gravitational binding energy , one gets the estimate . For a typical neutron star with and , this yields for the maximum value of the magnetic field strength G. This magnitude can be expected in the interior of a magnetar while recent observations report the surface values up to G, as inferred from the hydrogen spectral lines IShS .
In order to characterize spin ordering in neutron matter, it is convenient to introduce a neutron spin polarization parameter
Fig. 1 shows the dependence of the neutron spin polarization parameter from density, normalized to the nuclear saturation density , at zero temperature in the absence of the magnetic field. The spontaneous polarization develops at for the SLy4 interaction () and at for the SLy7 interaction (), that reflects the instability of neutron matter with the Skyrme interaction at such densities against spin fluctuations. Since the self-consistent equations at are invariant with respect to the global flip of neutron spins, we have two branches of solutions for the spin polarization parameter, (upper) and (lower) which differ only by sign, .
Fig. 2 shows the neutron spin polarization parameter as a function of density for a set of fixed values of the magnetic field. The branches of spontaneous polarization are modified by the magnetic field differently, since the self-consistent equations at lose the invariance with respect to the global flip of the spins. At nonvanishing , the lower branch , corresponding to the negative spin polarization, extends down to the very low densities. There are three characteristic density domains for this branch. At low densities , the absolute value of the spin polarization parameter increases with decreasing density. At intermediate densities , there is a plateau in the dependence, whose characteristic value depends on , e.g., at G. At densities , the magnitude of the spin polarization parameter increases with density, and neutrons become totally polarized at .
Note that the results in the low-density domain should be considered as a first approximation to the real complex picture, since, as discussed in detail in Ref. PG , the low density neutron-rich matter in -equilibrium possesses a frustrated state, ”nuclear pasta”, arising as a result of competition of Coulomb long-range interactions and nuclear short-range forces. In our case, where a pure neutron matter is considered, there is no mechanical instability due to the absence of the Coulomb interaction. However, the possibility of appearance of low-density nuclear magnetic pasta and its impact on the neutrino opacities in the protoneutron star early cooling stage should be explored in a more detailed analysis.
Let us consider the modification of the upper branch of spontaneous polarization at nonvanishing magnetic field. It is seen from Fig. 2 that now beginning from some threshold density the self-consistent equations at a given density have two positive solutions for the spin polarization parameter (apart from one negative solution). These solutions belong to two branches, and , characterized by different dependence from density. For the branch , the spin polarization parameter decreases with density and tends to zero value while for the branch it increases with density and is saturated. These branches appear step-wise at the same threshold density dependent on the magnetic field and being larger than the critical density of spontaneous spin instability in neutron matter. For example, for SLy7 interaction, at G, and at G. The magnetic field, due to the negative value of the neutron magnetic moment, tends to orient the neutron spins opposite to the magnetic field direction. As a result, the spin polarization parameter for the branches , with the positive spin polarization is smaller than that for the branch of spontaneous polarization , and, vice versa, the magnitude of the spin polarization parameter for the branch with the negative spin polarization is larger than the corresponding value for the branch of spontaneous polarization . Note that the impact of even such strong magnetic field as G is small: The spin polarization parameter for all three branches - is either close to zero, or close to its value in the state with spontaneous polarization, which is governed by the spin-dependent medium correlations.
Thus, at densities larger than , we have three branches of solutions: one of them, , with the negative spin polarization and two others, and , with the positive polarization. In order to clarify, which branch is thermodynamically preferable, we should compare the corresponding free energies. Fig. 3 shows the energy per neutron as a function of density at and G for these three branches, compared with the energy per neutron for a spontaneously polarized state [the branches ]. It is seen that the state with the majority of neutron spins oriented opposite to the direction of the magnetic field [the branch ] has a lowest energy. This result is intuitively clear, since magnetic field tends to direct the neutron spins opposite to , as mentioned earlier. However, the state, described by the branch with the positive spin polarization, has the energy very close to that of the thermodynamically stable state. This means that despite the presence of a strong magnetic field G, the state with the majority of neutron spins directed along the magnetic field can be realized as a metastable state in the dense core of a neutron star in the model consideration with the Skyrme effective interaction. In this scenario, since such states exist only at densities , under decreasing density (going from the interior to the outer regions of a magnetar) a metastable state with the positive spin polarization at the threshold density changes into a thermodynamically stable state with the negative spin polarization.
At this point, note some important differences between the results in our study and those obtained in Ref. PG . First, in the study PG of neutron matter at a strong magnetic field only one branch of solutions for the spin polarization parameter was found in the model with the Skyrme interaction (for the same SLy4 and SLy7 parametrizations). However, in fact, we have seen that the degenerate branches of spontaneous polarization (at zero magnetic field) with the positive and negative spin polarization are modified differently by the magnetic field, and, as a result, in the Skyrme model, in general, there are three different branches of solutions of the self-consistent equations at nonvanishing magnetic field. Besides, the only branch considered in Ref. PG and corresponding to our thermodynamically stable branch , is characterized by the positive spin polarization, contrary to our result with . This disagreement is explained by the incorrect sign before the term with the magnetic field in the equation for the quasiparticle spectrum in Ref. PG (analogous to Eq. (19) in our case). Clearly, in the equilibrium configuration the majority of neutron spins are aligned opposite to the magnetic field.
Fig. 4 shows the spin polarization parameter as a function of the magnetic field strength at zero temperature for different branches - of solutions of the self-consistent equations at compared with that for the branch at . It is seen that up to the field strengths G, the influence of the magnetic field is rather marginal. For the branches and , the magnitude of the spin polarization parameter increases with the field strength while for the it decreases. Interestingly, as is clearly seen from the top panel for the SLy4 interaction, at the given density, there exists some maximum magnetic field strength at which the branches and converge and do not continue at .
Fig. 5 shows the energy of neutron matter per particle as a function of the magnetic field strength at under the same assumptions as in Fig. 4. It is seen that the state with the negative spin polarization [branch ] becomes more preferable with increasing the magnetic field although the total effect of changing the magnetic field strength by two orders of magnitude on the energy corresponding to all three branches - remains small. It is also seen that the increase of the density by a factor of two leads to the increase in the energy per neutron roughly by a factor of three reflecting the fact that the medium correlations play more important role in building the energetics of the system than the impact of a strong magnetic field.
We have considered spin polarized states in neutron matter at a strong magnetic field in the model with the Skyrme effective NN interaction (SLy4, SLy7 parametrizations). The self-consistent equations for the spin polarization parameter and chemical potential of neutrons have been obtained and analyzed at zero temperature. It has been shown that the thermodynamically stable branch of solutions for the spin polarization parameter as a function of density corresponds to the case when the majority of neutron spins are oriented opposite to the direction of the magnetic field (negative spin polarization). This branch extends from the very low densities to the high density region where the spin polarization parameter is saturated, and, respectively, neutrons become totally spin polarized. Besides, beginning from some threshold density being dependent on the magnetic field strength the self-consistent equations have also two other branches (upper and lower) of solutions for the spin polarization parameter corresponding to the case when the majority of neutron spins are oriented along the magnetic field (positive spin polarization). For example, for SLy7 interaction, at G, and at G. The spin polarization parameter along the upper branch increases with density and is saturated, while along the lower branch it decreases and vanishes. The free energy corresponding to the upper branch turns out to be very close to the free energy corresponding to the thermodynamically preferable branch with the negative spin polarization. As a consequence, at a strong magnetic field, the state with the positive spin polarization can be realized as a metastable state at the high density region in neutron matter which under decreasing density (going from the interior to the outer regions of a magnetar) at the threshold density changes into a thermodynamically stable state with the negative spin polarization.
In this study, we have considered the zero temperature case, but as was shown in Ref. PG , the influence of finite temperatures on spin polarization remains moderate in the Skyrme model, at least, up to the temperatures relevant for protoneutron stars, and, hence, one can expect that the considered scenario will be preserved at finite temperatures as well. The possible existence of a metastable state with positive spin polarization will affect the neutrino opacities of a neutron star matter in a strong magnetic field, and, hence, will lead to the change of cooling rates of a neutron star compared to cooling rates in the scenario with the majority of neutron spins oriented opposite to the magnetic field PG2 .
The calculations of the neutron spin polarization parameter and energy per neutron show that the influence of the magnetic field remains small at the field strengths up to G. Note that in Ref. PG the consideration also has been done for the Gogny effective NN interaction (D1S, D1P parametrizations) up to densities . Since for the D1S parametrization there is no spontaneous ferromagnetic transition in neutron matter for all relevant densities, and for the D1P parametrization this transition occurs at the density larger than LVRP , no sign of a ferromagnetic transition at a strong magnetic field was found in Ref. PG up to densities for these Gogny forces. According to our consideration, one can expect that the metastable states with the positive spin polarization in neutron matter at a strong magnetic field could appear at densities larger than for the D1P parametrization while the scenario with the only branch of solutions corresponding to the negative spin polarization would be realized for the D1S force.
It is worthy to note also that in the given research a neutron star matter was approximated by pure neutron matter. This approximation allows one to get the qualitative description of the spin polarization phenomena and should be considered as a first step towards a more realistic description of neutron stars taking into account a finite fraction of protons with the charge neutrality and beta equilibrium conditions. In particular, some admixture of protons can affect the onset densities of enhanced polarization in a neutron star matter with the Skyrme interaction.
J.Y. was supported by grant R32-2008-000-10130-0 from WCU project of MEST and NRF through Ewha Womans University.
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